Reflexivity

A
relation, Rxy, (that is, the relation expressed by "Rxy") is
reflexive in a domain just if there is no dot in its graph without
a loop – i.e. just if everything in the domain bears the relation to itself.
Expressed formally,

Rxy
is reflexive just if "xRxx

Examples: "x=y"
is reflexive in every domain; "x is the same age as y" is reflexive
in the domain of living things, but not, however, in a domain which contains
numbers – since, presumably, the number 5 is not the same age as itself,
since it isn't any age at all.

So notice that
a relation may have a property in one domain which it lacks in others.

Rxy
is irreflexive in a domain just if no dot in its graph has a loop.

That is:

Rxy
is irreflexive just if "x¬Rxx

Examples: "x
is older than y"; "x>y".

Rxy
is non-reflexive just if it is neither reflexive nor
irreflexive – i.e. at least one of the dots in its graph has a loop and
at least one does not. That is:

Rxy
is non-reflexive just if [$xRxxÙ$x¬Rxx].

(We could have
said: just if [¬"xRxxÙ¬"x¬Rxx]. Remember that "¬"xj"
is equivalent to "$x¬j".)

Notice that in
the empty domain a relation will be both reflexive and
irreflexive: since there will be no dots at all, it follows that there
will be no dots with loops (so it is irreflexive) and no dots without
loops (so it will be reflexive).

Remember also that
in the empty domain anything beginning """ will be true (as long as it contains
no names, in which case the formula can't have an interpretation with
an empty domain).